Learning Outcomes
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- Define the identity and zero properties of exponents and logarithms.
- Define the inverse property of exponents and logarithms.
- Define the one-to-one properties of exponents and logarithms.
- Define the product and quotient properties of exponents and logarithms.
When you learned how to solve linear equations, you were likely introduced to the properties of real numbers. These properties help us know what the rules are for isolating and combining numbers and variables. For example, it is advantageous to know that multiplication and division “undo” each other when you want to solve an equation for a variable that is multiplied by a number. You may have also been introduced to properties and rules for writing and using exponents. In this section, we will show you how the properties and rules for logarithms compare to those for exponents.
Recall that we can express the relationship between logarithmic form and its corresponding exponential form as follows:
[latex]\log_b\left(x\right)=y\ \Leftrightarrow \ {b}^{y}=x,\ \text{}b>0,\ b\ne 1[/latex]
That is, to say that the logarithm to base b of [latex]x[/latex] is [latex]y[/latex] is equivalent to saying that [latex]y[/latex] is the exponent on the base b that produces [latex]x[/latex].
Note that the base b is always a positive number other than [latex]1[/latex] and that the logarithmic and exponential forms “undo” each other. This means that logarithms have properties similar to exponents. Let’s compare several such properties side by side.
Identity and Zero Properties of Exponents and Logarithms
Zero Exponent Rule for Logarithms and Exponentials
Recall that for exponents,
[latex]b^0=1[/latex].
Using the definition of a logarithm, we can rewrite that statement in logarithm form:
[latex]\log_b1=0[/latex]
Identity Exponent Rule for Logarithms and Exponentials
Recall that for exponents,
[latex]b^1 = b[/latex]
Using the definition of a logarithm, we can rewrite that statement in logarithm form:
[latex]\log_bb=1[/latex]
Example
Use the the fact that exponentials and logarithms are inverses to prove the zero and identity exponent rule for the following:
1. [latex]{\mathrm{log}}_{5}1=0[/latex]
2. [latex]{\mathrm{log}}_{5}5=1[/latex]
Inverse Property of Logarithms and Exponents
Exponential and logarithmic functions are inverses of each other, and we can take advantage of this to evaluate and solve expressions and equations involving logarithms and exponentials. The inverse property of logarithms and exponentials gives us an explicit way to rewrite an exponential as a logarithm or a logarithm as an exponential.
Inverse Property of Logarithms and Exponentials
Recall that we can use the definition of a logarithm to rewrite it as an exponent. That is
[latex]\log_bx=y \ \Leftrightarrow b^y=x[/latex].
By extension,
[latex]\log_bb^x=x \qquad[/latex] and [latex]\qquad b^{\log_bx}=x[/latex].
Example
Evaluate:
1.[latex]\mathrm{log}\left(100\right)[/latex]
2.[latex]{e}^{\mathrm{ln}\left(7\right)}[/latex]
One-to-One Property of Logarithms and Exponents
Consider the mathematical statement [latex]2^x=2^y[/latex]. The only possible way this could be a true statement is if the variables [latex]x[/latex] and [latex]y[/latex] are equivalent. This statement is an example of the one-to-one property of exponents. More formally, it states
[latex]a^m = a^n \Rightarrow m=n[/latex].
In more general terms, if like bases raised to perhaps different exponents are given to be equivalent, then the exponents themselves must also be equivalent.
The same idea applies to logarithms. If two logarithms of like base but with perhaps different arguments are given to be equivalent, then the arguments themselves must also be equivalent. More formally, the one-to-one property of logarithms states
[latex]\log_b(M) = \log_b(N) \Rightarrow M = N[/latex].
Example
Solve the equation [latex]{\mathrm{log}}_{3}\left(3x\right)={\mathrm{log}}_{3}\left(2x+5\right)[/latex] for [latex]x[/latex].
The Product and Quotient Rules for Logarithms
Recall that we use the product rule of exponents to combine the product of like bases raised to exponents by adding the exponents:
[latex]{x}^{a}{x}^{b}={x}^{a+b}[/latex].
We have a similar property for logarithms, called the product rule for logarithms, which says that the logarithm of a product is equal to a sum of logarithms. Because logs are exponents, and we multiply like bases, we can add the exponents. We will use the inverse property to derive the product rule below.
The Product Rule for Logarithms
The product rule for logarithms can be used to simplify a logarithm of a product by rewriting it as a sum of individual logarithms.
[latex]{\mathrm{log}}_{b}\left(MN\right)={\mathrm{log}}_{b}\left(M\right)+{\mathrm{log}}_{b}\left(N\right)[/latex]
Example
Using the product rule for logarithms, rewrite the logarithm of a product as the sum of logarithms of its factors.
[latex]{\mathrm{log}}_{b}\left(wxyz\right)[/latex]
For quotients, we have a similar rule for logarithms. Recall that we use the quotient rule of exponents to simplify division of like bases raised to powers by subtracting the exponents:
[latex]\dfrac{x^a}{x^b}={x}^{a-b}[/latex].
The quotient rule for logarithms says that the logarithm of a quotient is equal to a difference of logarithms. Just as with the product rule, we can use the inverse property to derive the quotient rule.
The Quotient Rule for Logarithms
The quotient rule for logarithms can be used to simplify a logarithm or a quotient by rewriting it as the difference of individual logarithms.
Example
Expand the following expression using the quotient rule for logarithms.
[latex]\mathrm{log}\left(\dfrac{2{x}^{2}+6x}{3x+9}\right)[/latex]
Later in the module, you’ll extend these properties and rules to more complicated examples and use them to write compact logarithms in expanded form and contract those already expanded. You’ll also obtain the power rule for logarithms, which will be particularly helpful when writing and solving exponential models of real-world situations.
